Largest similar copies of convex polygons amidst polygonal obstacles

TL;DR

This paper presents an improved algorithm for finding the largest similar copy of a convex polygon within a polygonal domain with obstacles, advancing the computational efficiency of a problem studied for over 25 years.

Contribution

The authors develop a faster algorithm with complexity $O(k^2n^2 ext{lambda}_4(k) ext{log}n)$ for the largest similar copy problem, improving over previous methods.

Findings

New algorithm with improved time complexity

Progress over 25 years of prior research

Enhanced computational approach for polygon placement

Abstract

Given a convex polygon $P$ with $k$ vertices and a polygonal domain $Q$ consisting of polygonal obstacles with total size $n$ in the plane, we study the optimization problem of finding a largest similar copy of $P$ that can be placed in $Q$ without intersecting the obstacles. We improve the time complexity for solving the problem to $O(k^2n^2\lambda_4(k)\log{n})$. This is progress of improving the previously best known results by Chew and Kedem [SoCG89, CGTA93] and Sharir and Toledo [SoCG91, CGTA94] on the problem in more than 25 years.